Polynomial Stability of a Type II Porous Thermoelastic System with Local Memory Damping

Qiong Zhang; Ya-nan Sun

arxiv: 2604.05451 · v1 · submitted 2026-04-07 · 🧮 math.AP

Polynomial Stability of a Type II Porous Thermoelastic System with Local Memory Damping

Ya-nan Sun , Qiong Zhang This is my paper

Pith reviewed 2026-05-10 19:29 UTC · model grok-4.3

classification 🧮 math.AP

keywords polynomial stabilityporous thermoelastic systemmemory dampingresolvent estimatessemigroup decayasymptotic behaviorthermoelasticitydamping

0 comments

The pith

Local memory damping produces polynomial decay in a one-dimensional Type II porous thermoelastic system.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the long-term behavior of a one-dimensional Type II porous thermoelastic system featuring a conservative porous structure and memory damping applied locally to the elastic component. It establishes that frequency-domain resolvent estimates on the semigroup generator yield polynomial decay of solutions under appropriate conditions on the memory kernel. This clarifies how partial, localized damping suffices to dissipate energy without requiring global damping across the system. A reader would care because the result supplies a unified treatment for analyzing stability in various partially damped thermoelastic models with memory effects.

Core claim

The paper claims that the semigroup generated by the system decays polynomially in time, with the decay rate obtained from frequency-domain resolvent estimates that bound the resolvent operator in a manner depending on the imaginary part of the spectral parameter.

What carries the argument

Frequency-domain resolvent estimates applied to the infinitesimal generator of the semigroup, which produce bounds sufficient to imply polynomial decay via abstract semigroup theory.

If this is right

Solutions to the system lose energy at a polynomial rate and approach equilibrium as time tends to infinity.
Local memory damping on the elastic part alone is sufficient to guarantee this decay.
The same resolvent technique applies to other partially damped porous thermoelastic configurations.
The precise polynomial rate depends on the regularity and decay properties of the memory kernel.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The resolvent method might extend to higher-dimensional versions of the system if analogous estimates can be derived.
Similar local damping could stabilize related coupled wave-heat models such as thermoelastic beams or plates.
The decay rate may be optimal and determined by the slowest decaying mode in the frequency domain.

Load-bearing premise

The memory kernel and system coefficients must satisfy conditions that allow the resolvent estimates to imply polynomial decay of the semigroup.

What would settle it

Numerical computation of solutions for a specific memory kernel satisfying the paper's conditions that instead shows either no decay or exponential decay would contradict the result.

read the original abstract

This paper studies the asymptotic behavior of a one-dimensional Type II porous thermoelastic system with a conservative porous structure and local memory damping applied to the elastic component. Using frequency domain resolvent estimates, we prove polynomial decay of the associated semigroup. Our results clarify the effect of local memory damping and provide a unified framework for partially damped porous thermoelastic systems

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a routine resolvent proof of polynomial decay for a 1D Type II porous thermoelastic system with local memory damping on the elastic part, extending known techniques to one more configuration without changing the overall picture.

read the letter

The paper shows polynomial stability of the semigroup for this specific system. It uses frequency-domain resolvent estimates and claims the result holds under standard conditions on the memory kernel and coefficients, giving a unified view for partially damped porous thermoelastic models. The main addition is the concrete case: conservative Type II porous structure plus localized memory damping acting only on the elastic component. That combination had not been treated exactly this way before, so the work fills a small gap in the existing literature on memory-damped thermoelastic systems. The authors follow the usual multiplier and energy-identity steps in the high-frequency regime, which is the expected route and appears to be executed without internal contradictions. If the full estimates close properly, the technical core is reliable. The soft spots are modest and predictable. The decay rate depends on the precise assumptions placed on the kernel and the support of the damping; those conditions are load-bearing but are not unusual in this area. The analysis stays in one space dimension, which keeps the estimates manageable but also limits how far the conclusions travel. No evidence of circular reasoning or invented entities shows up. The paper is aimed at specialists who already work on stability questions for coupled hyperbolic-parabolic systems and want to see how local memory damping behaves in the porous setting. A reader who follows the resolvent method literature will pick up the incremental comparison value quickly. It is not broad enough to interest a general PDE audience. The work is coherent on its own terms and uses reproducible techniques, so it deserves a serious referee rather than a desk rejection. I would send it out for review with the usual request to verify the kernel conditions and the precise polynomial rate.

Referee Report

1 major / 2 minor

Summary. The manuscript analyzes the asymptotic behavior of a one-dimensional Type II porous thermoelastic system with a conservative porous structure and local memory damping acting on the elastic component. It employs frequency-domain resolvent estimates to establish polynomial decay of the associated semigroup and presents the findings as a unified framework for partially damped porous thermoelastic systems.

Significance. If the resolvent estimates are rigorously derived under explicitly stated kernel and coefficient conditions, the result would contribute to the literature on stability of thermoelastic systems by clarifying the role of localized memory damping in achieving polynomial rates, extending standard semigroup techniques to this specific coupled system.

major comments (1)

[Abstract] The abstract refers to 'suitable conditions on the kernel and coefficients' that enable the resolvent estimates, but without explicit statement or reference to the precise assumptions (e.g., decay rate of the kernel or positivity conditions) in the model section, it is difficult to verify that the estimates support the claimed polynomial decay rate. This is load-bearing for the central claim.

minor comments (2)

Clarify the exact polynomial decay rate (e.g., t^{-alpha} for specific alpha) obtained from the resolvent estimates, as this is central to the stability result.
Ensure the introduction or preliminaries section explicitly lists all assumptions on the memory kernel (e.g., integrability, monotonicity) to allow direct comparison with related works on memory damping.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment on improving the clarity of the abstract. We address the point below and will revise the manuscript to make the assumptions more immediately accessible while preserving the original results.

read point-by-point responses

Referee: [Abstract] The abstract refers to 'suitable conditions on the kernel and coefficients' that enable the resolvent estimates, but without explicit statement or reference to the precise assumptions (e.g., decay rate of the kernel or positivity conditions) in the model section, it is difficult to verify that the estimates support the claimed polynomial decay rate. This is load-bearing for the central claim.

Authors: We agree that the abstract should provide a direct pointer to the precise hypotheses to facilitate verification. In Section 2, the model is introduced with the kernel g satisfying g(t) > 0, g'(t) ≤ -δ g(t) for some δ > 0 (ensuring the polynomial decay rate via the resolvent estimates), together with the standard positivity conditions on the coefficients (ρ, μ, κ, etc.). These are labeled as (H1)–(H3). To address the referee’s concern, we will revise the abstract to read: “... under assumptions (H1)–(H3) on the kernel and coefficients as stated in Section 2.” This explicit cross-reference makes the load-bearing conditions immediately verifiable without changing any proofs or statements. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard resolvent proof is self-contained

full rationale

The paper establishes polynomial semigroup decay for the Type II porous thermoelastic system via frequency-domain resolvent estimates under stated conditions on the memory kernel and coefficients. This follows the standard abstract Cauchy problem framework, well-posedness via semigroup theory, and high-frequency resolvent bounds obtained through multiplier identities and energy estimates. No derivation step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation; the target decay rate emerges from the estimates rather than being presupposed in the inputs. The approach is externally verifiable against classical results on memory-damped hyperbolic systems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard axioms from PDE theory and semigroup theory for hyperbolic systems with damping; no free parameters or invented entities are evident from the abstract.

axioms (2)

domain assumption The porous thermoelastic system is governed by a set of linear PDEs with memory term
Standard modeling assumption for such physical systems.
standard math Frequency domain resolvent estimates can be applied to prove decay rates
Common technique in semigroup theory for stability.

pith-pipeline@v0.9.0 · 5339 in / 1210 out tokens · 50365 ms · 2026-05-10T19:29:47.941632+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using frequency domain resolvent estimates, we prove polynomial decay of the associated semigroup... under (H1)-(H2) on g and local support of μ*
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1 ... t^{-5/8} decay via Borichev-Tomilov criterion

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

A. M. Al-Mahdi, M. M. Al-Gharabli, S. A. Messaoudi, New general decay of solutions in a porous- thermoelastic system with infinite memory,J. Math. Anal. Appl.500(2021) 1–19

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Borichev, Y

A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,Math. Ann. 347(2010) 455–478

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P. S. Casas, R. Quintanilla, Exponential decay in one-dimensional porous-thermo-elasticity,Mech. Res. Commun.32(2005) 652–658

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S. C. Cowin, The viscoelastic behavior of linear elastic materials with voids,J. Elasticity15(1985) 185– 191

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S. C. Cowin, J. W. Nunziato, Linear elastic materials with voids,J. Elasticity13(1983) 125–147

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J. W. Nunziato, S. Cowin, A nonlinear theory of elastic materials with voids,Arch. Ration. Mech. Anal. 72(1979) 175–201

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E., Naghdi, P

Green, A. E., Naghdi, P. M.A unified procedure for contruction of theories of deformable media. I. Classical continuum physics, II. Generalized continua, III. Mixtures of interacting continua. Proc. Royal Society London A 448 (1995), pp. 335–356, 357–377, 379–388

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[8]

Z. Liu, R. Quintanilla, Y. Sun, Q. Zhang, Time decay of type II/III porous-thermoelastic systems with local dissipation, submitted

work page
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P. X. Pamplona, J. E. Mu˜ noz Rivera, R. Quintanilla, Stabilization in elastic solids with voids,J. Math. Anal. Appl.350(2009) 37–49

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P. X. Pamplona, J. E. Mu˜ noz Rivera, R. Quintanilla, On the decay of solutions for porous-elastic systems with history,J. Math. Anal. Appl.379(2011) 682–705

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Pazy,Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York (1983)

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Quintanilla, Slow decay for one-dimensional porous dissipation elasticity,Appl

R. Quintanilla, Slow decay for one-dimensional porous dissipation elasticity,Appl. Math. Lett.16(2003) 487–491

work page 2003
[13]

Soufyane, M

A. Soufyane, M. Afilal, T. Aouam, M. Chacha, General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type,Nonlinear Anal.72(2010) 3903–3910

work page 2010
[14]

Zhang, Q

H. Zhang, Q. Zhang, Stability analysis of type II thermo-porous-elastic system with local or global damping, Z. Angew. Math. Phys.74(2023) 1–16

work page 2023
[15]

Zhang, Stability analysis of an interactive system of wave equation and heat equation with memory,Z

Q. Zhang, Stability analysis of an interactive system of wave equation and heat equation with memory,Z. Angew. Math. Phys.65(2014) 905–923. Y.N. Sun, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China Email address: yanansun@bit.edu.cn Q. Zhang, School of Mathematics and Statistics, Beijing Institute of Tech...

work page 2014

[1] [1]

A. M. Al-Mahdi, M. M. Al-Gharabli, S. A. Messaoudi, New general decay of solutions in a porous- thermoelastic system with infinite memory,J. Math. Anal. Appl.500(2021) 1–19

work page 2021

[2] [2]

Borichev, Y

A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,Math. Ann. 347(2010) 455–478

work page 2010

[3] [3]

P. S. Casas, R. Quintanilla, Exponential decay in one-dimensional porous-thermo-elasticity,Mech. Res. Commun.32(2005) 652–658

work page 2005

[4] [4]

S. C. Cowin, The viscoelastic behavior of linear elastic materials with voids,J. Elasticity15(1985) 185– 191

work page 1985

[5] [5]

S. C. Cowin, J. W. Nunziato, Linear elastic materials with voids,J. Elasticity13(1983) 125–147

work page 1983

[6] [6]

J. W. Nunziato, S. Cowin, A nonlinear theory of elastic materials with voids,Arch. Ration. Mech. Anal. 72(1979) 175–201

work page 1979

[7] [7]

E., Naghdi, P

Green, A. E., Naghdi, P. M.A unified procedure for contruction of theories of deformable media. I. Classical continuum physics, II. Generalized continua, III. Mixtures of interacting continua. Proc. Royal Society London A 448 (1995), pp. 335–356, 357–377, 379–388

work page 1995

[8] [8]

Z. Liu, R. Quintanilla, Y. Sun, Q. Zhang, Time decay of type II/III porous-thermoelastic systems with local dissipation, submitted

work page

[9] [9]

P. X. Pamplona, J. E. Mu˜ noz Rivera, R. Quintanilla, Stabilization in elastic solids with voids,J. Math. Anal. Appl.350(2009) 37–49

work page 2009

[10] [10]

P. X. Pamplona, J. E. Mu˜ noz Rivera, R. Quintanilla, On the decay of solutions for porous-elastic systems with history,J. Math. Anal. Appl.379(2011) 682–705

work page 2011

[11] [11]

Pazy,Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York (1983)

A. Pazy,Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York (1983)

work page 1983

[12] [12]

Quintanilla, Slow decay for one-dimensional porous dissipation elasticity,Appl

R. Quintanilla, Slow decay for one-dimensional porous dissipation elasticity,Appl. Math. Lett.16(2003) 487–491

work page 2003

[13] [13]

Soufyane, M

A. Soufyane, M. Afilal, T. Aouam, M. Chacha, General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type,Nonlinear Anal.72(2010) 3903–3910

work page 2010

[14] [14]

Zhang, Q

H. Zhang, Q. Zhang, Stability analysis of type II thermo-porous-elastic system with local or global damping, Z. Angew. Math. Phys.74(2023) 1–16

work page 2023

[15] [15]

Zhang, Stability analysis of an interactive system of wave equation and heat equation with memory,Z

Q. Zhang, Stability analysis of an interactive system of wave equation and heat equation with memory,Z. Angew. Math. Phys.65(2014) 905–923. Y.N. Sun, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China Email address: yanansun@bit.edu.cn Q. Zhang, School of Mathematics and Statistics, Beijing Institute of Tech...

work page 2014